Morava Research in the Mathematical Sciences (2015) 2:i DOI 10.1186/s40687-015-0028-7

0 Research in

the Mathematical Sciences

a SpringerOpen Journal

RESEARCH Open Access

Homotopy-theoretically enriched categories of noncommutative motives

Jack Morava

Correspondence:

jack@math.jhu.edu

The Johns Hopkins University, 3400

North Charles Street, Baltimore,

Maryland 21218, USA

Abstract

Waldhausen's K-theory of the sphere spectrum (closely related to the algebraic K-theory of the integers) is naturally augmented as an 5°-algebra, and so has a Koszul dual. Classic work of Deligne and Goncharov implies an identification of the rationalization of this (covariant) dual with the Hopf algebra of functions on the motivic group for their category of mixed Tate motives over Z. This paper argues that the rationalizations of categories of noncommutative motives defined recently by Blumberg, Gepner, and Tabuada consequently have natural enrichments, with morphism objects in the derived category of mixed Tate motives over Z. We suggest that homotopic descent theory lifts this structure to define a category of motives defined not over Z but over the sphere ring-spectrum 5°. 1991 Mathematics subject classification: 11G, 19F,57R,81T

1 Introduction

1.1 Building on earlier work going back at least three decades [26], Deligne and Goncharov have defined a Q-linear Abelian rigid tensor category of mixed Tate motives over the integers of a number field: in particular, the category MTq(Z) of such motives over the rational integers. Its generators are tensor powers Q(n) = Q(1)®n of a Tate object Q(1), inverse to the Lefschetz hyperplane motive (which can be regarded as a degree two shift of the complex

0 ^ P1 ^ P0 ^ 0

ringer

in Voevodsky's derived category). We argue here that these objects are analogous to the (even-dimensional) cells of stable homotopy theory: in that, for example, the image

Pn = Q(0) e---e Q(-n)

of projective space in this category splits as a sum of terms resembling Lefschetz's hyperplane sections.

Deligne and Goncharov's definition ([27], § 1.6) depends on the validity of the Beilinson-Soule vanishing conjecture for number fields, which implies that their category MTq(Z) can be characterized by a very simple spectral sequence with £2-term

ExtMr(Q(0), Q(n)) ^ K(Zhn-* ® Q

© 2015 Morava. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http:// creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

equal to zero if * > lor when * = 0, n = 0. Borel's theory of regulators [19,29] identifies the nonvanishing groups

Kik+iiZ) ® Q c R

with the subgroup of rational multiples of the conjecturally transcendental values f(1 + 2k) of the Riemann zeta function at odd positive integers.

To a homotopy theorist, this is strikingly reminiscent of Atiyah's interpretation of Adams' work on Whitehead's homomorphism

Jn-i : KOn(*) = nn-iO ^ lim nm+n-i(Sm) = i(*) ,

i.e., the effect on homotopy groups of the monoid map O = lim O(m) ^ lim ^m-1Sm-1 := Q(S0) .

m^TO m^TO

The image of a Bott generator

KO4k(*) = Z ^ image J4k-i = (2f(1 - 2k) • Z)/Z c Q/Z

under this homomorphism can be identified with the (rational) value of the zeta function at an odd negative integer.

Here is a quick sketch of this argument: A stable real vector bundle over S4k is classified by its equatorial twist

5 : S4k-1-^ O-* Q(S0) ,

which defines a stable cofibration

S4k-1 s0-" Cof ¿5-S4k..........> ••• .

Adams' e-invariant is the class of the resulting sequence

0-^ KO(S4k)-KO(Cof ¿5)-f KO(S0)-^ 0

in a group

ExtAdams (KO(S0), KO(S4k))

of extensions of modules over the stable KO-cohomology operations. [KO is contravari-ant, while motives are covariant, making KO of a sphere analogous to a Tate object.] After profinite completion [4], these cohomology operations become an action of the group Zx of units in the profinite integers - in fact the action, through its abelianization, of the absolute Galois group of Q - and the resulting group of extensions can be calculated in terms of generalized Galois cohomology as

Ext|x (Z(0), Z(2k)) = Hi (Zx, Z(2k))

(where u e Zx acts on Z(n) as multiplication by un). At an odd primep, H (Z£, Zp(2k)) is zero unless 2k = (p — 1)ko, when the group is cyclic ofp-order vp(k0) +1. By congruences of von Staudt and Clausen, this is the p-order of the Bernoulli quotient

B2k Ik e

a global argument over Q (i.e., using the Chern character ([1], § 7.1b)) refines this to a homomorphism

H(Zx, Z(2k)) ^ Q/Z

which sends a generator ofKO(S4k) to the class of §f(1 - 2k). See also ([26], § 3.5).

1.2 This paper proposes an analog of the theory of mixed Tate motives in the world of stable homotopy theory, based on Bokstedt's theorem ([16], or more recently [14]) that the morphism

K(S0) ^ K(Z)

of ring-spectra (induced by the Hurewicz morphism

[1: S0 ^ HZ] e H0(S0, Z))

becomes an isomorphism after tensoring with Q. At this point, odd zeta-values enter differential topology [43]. To be more precise, we argue that (unlike K(Z)) K(S°) is naturally augmented as a ring-spectrum over S0, via the Dennis trace

tro : K(S0) ^ THH(S0) - S0

[70]. Current work [38,64] on descent in homotopy theory suggests the category of comodule spectra over the covariant Koszul dual

S0 4(S0) S0 := K(S0)

of K(S°) (or perhaps more conventionally, the category of module spectra over RHomK(S0)(S0, S0))

as a natural candidate for a homotopy-theoretic analog of MTq(Z). This paper attempts to make this plausible after tensoring K(S°) with the rational field Q.

1.3 Organization Koszul duality is a central concern of this paper; in its most classical form, it relates (graded) exterior and symmetric Hopf algebras. The first section below observes that the Hopf algebra of quasisymmetric functions is similarly related to a certain odd-degree square-zero augmented algebra. Stating this precisely (i.e., over Z) requires comparison of the classical shuffle product ([31], Ch II) with the less familiar quasi-shuffle product [40,50]. I am especially indebted to Andrew Baker and Birgit Richter for explaining this to me.

The next section defines topologically motivated generators (quite different from those of Borel) for K*(S0) < Q. Work of Hatcher ([36], § 6.4), Waldhausen [69], and Bokstedt on pseudo-isotopy theory has been refined by Rognes [63] to construct an infinite-loop map

a : B(F/O) ^ Wh(*) (c K(S0))

(F being the monoid of homotopy self-equivalences of the stable sphere) which is a rational equivalence. This leads to the definition of a homotopy equivalence

w : (S0 v ZkO) < Q ^ K(S0) < Q

of ring-spectra, with a square-zero extension of the rational sphere spectrum on the left, which can then be compared with Borel's calculations. Some of the work of Deligne and Goncharov is then summarized to construct a lift of this rational isomorphism to

an equivalence between the algebra of functions on the motivic group of the

Tannakian category MTq(Z) and the covariant Koszul dual K(S0)* ® Q.

The final section is devoted to applications: in particular to the 'decategorifica-tion' ([18], [49], § 4) of two-categories of 'big' (noncommutative) motives constructed by Blumberg, Gepner, and Tabuada [10,11], and to work of Kitchloo [46] on categories of symplectic analogs of motives. The objects in the categories of 'big' motives are themselves small stable ^-categories, with stable ^-categories of suitably exact functors between them as morphism objects. The (Waldhausen) K-theory spectra of these morphism categories define new categories enriched over the homotopy category of K(S0)-module spectra ([11], Corollary 4.13), having the original small stable categories as objects.

'Rationalizing' (tensoring the morphism objects in these homotopy categories with Q) defines categories enriched over K*(S°) <g> Q-modules, to which the Koszul duality machinery developed here can be applied. Under suitable finiteness hypotheses, this constructs categories of noncommutative motives enriched over the derived category Db(MTQ (Z) of classical mixed Tate motives.

2 Quasisymmetric functions and Koszul duality

2.1 The fundamental example underlying this paper could well have appeared in Tate's 1957 work [67] on the homology of local rings; but as far as I know, it is not in the literature, so I will begin with it: Let £* := £* [ e2k+\ | k > 0] be the primitively generated graded-commutative Hopf algebra over Z with one generator in each odd degree, and let

: £* ^ £* /£+ = Z

be the quotient by its ideal £+ of positive-degree elements; then

Tor£(Z, Z) = P[X2(k+1) I k > 0] (:= Symm*)

is a graded-commutative Hopf algebra with one generator in each even degree, canoni-cally isomorphic to the classical algebra of symmetric functions with coproduct

Ax(t) = x(t) <g> x(t)

(x(t) = ^ X2ktk, X0 := 1). k>0

This is an instance of a very general principle: if A* ^ k is an augmented commutative graded algebra (assuming for simplicity that k is a field), then

k ®A* k = TorA(k, k) := At

is an augmented, graded-commutative Hopf algebra, with

ExtA(k,k) := RHomA(k,k)

as its graded dual ([22], XVI § 6). More generally,

(A - Mod) 3 M ^ TorA (M, k) := Mt

extends this construction to a functor taking values in a category of graded A*-comodules. This fascinated John Moore [41,60], and its implications have become quite important in representation theory [7,8]; more recently, the whole subject has been vastly generalized by the work of Lurie.

2.2.1 For our purposes it is the quotient

VE : E* ^ E*/(E+)2 := f*

of the exterior algebra above, by the ideal generated by products of positive-degree elements, which is relevant. This quotient is the square-zero extension

E* = Z 0 E+ = Z 0 fek+1 | k > 0}

of Z by a graded module with one generator in each odd degree.

Proposition. After tensoring with Q, the induced homomorphism vE : Torf (Z, Z) = Symm* ^ Torf (Z, Z) = QSymm*

of Hopf algebras is the inclusion of the graded algebra of rational symmetric functions into the algebra of rational quasi-symmetric functions, given the classical shuffle product ш.

Proof. In this case the classical bar resolution

B*(f/Z) = E* ®z (e„>oE+ [1]<n) Z

(of Z as an f*-module ([55] Ch X § 2.3, [31] Ch II, [24] § 2)) is, apart from the left-hand term, just the tensor algebra of the graded module E+[ 1] (obtained from E+ by shifting the degrees of its generators up by one), with algebra structure defined by the shuffle product; but I will defer discussing that till § 2.3 below. Since E* is a DGA with trivial differential and trivial product, the homology Tor^ (Z, Z) of the complex

Z < B*(E/Z) = 0n>o(E+ [ 1] )<n

with its resulting trivial differential is the algebra QSymm* on E+[ 1]. [Tate, by the way, worked with a commutative noetherian local ring

Фа : A ^ B = A/mA = k

and studied To^ (k, k), though not as a Hopf algebra; but in his calculations, he used what is visibly the resolution above.]

Remark. In fact under very general conditions ([38], § 3.1, [39], § 6.12), the bar construction associates to a morphism v : A ^ B of suitable monoid objects, a pullback functor

LV* : M*^ M* <a B*(A/B) = M*<LAB

from some (simplicial or derived) category of modules over Spec A to a similar category of modules over Spec B, cf § 2.4 below. Here B*(A/B) is a resolution of B as an A-module, corresponding to B(A,B, A) in ([32], Prop 7.5), cfalso ([15,31,33,51,53],...). In the example above, regarding E* as a DGA with trivial differential, we obtain a covariant functor from the bounded derived category of E*-modules to the bounded derived category of graded modules with a coaction of the classical Hopf algebra of symmetric functions. However, the algebra of symmetric functions is canonically self-dual over Z ([54], I § 4) and we can interpret this derived pullback as a functor to the bounded derived category of modules over the dual symmetric algebra.

2.2.2 In this paper I will follow K Hess ([38], § 2.2.23 - 2.2.28]): a morphism V : A ^ B

of monoids in a suitable (e.g., simplicially enriched ([38], § 3.16, § 5.3)) category of modules (perhaps over a differential graded algebra or a ring-spectrum) defines an A-bimodule bialgebra

BalaB := W(V)

(analogous to an algebraic topologist's Hopf algebroid, though in general without antipode). In her framework the construction above is a descent functor: its target has a natural ('Tannakian') enrichment ([37], § 5.3, [44]) or lift LV to a category of B-modules with compatible coaction by the descent coring W(v). Lurie's work (e.g., [52], § 7.13], [53] § 5.2.2 - 5.2.3) provides the natural context for such constructions.

2.2.3 Completing the argument requires clarifying relations between the shuffle product lli and the quasi-shuffle or 'stuffle' product □ . A shuffle of a pair r, s > 1of integers is a partition of the set {1,..., r + s} into disjoint subsets a\ < • • • < ar and bi < • • • < bs; such a shuffle defines a permutation

a(1,..., r + s) = (a\, ••• , ar, b1,..., bs) .

The shuffle product on the tensor algebra T'(V) of a module V is defined by

v\ • • • vr HI vr+1 • • • vr+s = ^ M1) • • • va(r+s) ,

with the sum taken over all shuffles of (r, s). The deconcatenation coproduct

A : T'(V) ^ T'(V) <g> T'(V)

sends v1 vr to the sum

v1 vr < 1 + v1 vi < vi+1 vr + 1 < v1 vr .

The algebra T'(V), with this (commutative but not cocommutative) Hopf structure, is sometimes called the cotensor (Hopf) algebra of V. The shuffle product is characterized by the identity

(v • x) lu (w •y) = v • (x lu (w •y)) + w • ((v • x) U1 y) ,

where v, w e V and x,y e T'(V). I will write QSymm* for the Hopf algebra Torf (Z, Z) of § 2.2.1, with lu as product.

The closely related Hopf algebra QSymm* of quasi-symmetric functions over Z, with the quasi-shuffle product □, is perhaps most efficiently defined as dual to the free graded associative Hopf algebra

NSymm* := Z^+d I k > 0>

of noncommutative symmetric functions ([6], [22], § 4.1.F, [36]), with coproduct

AZ(t) = Z(t) < Z(t)

(Z(t) = J2Z2ktk, Zc = 1).

More generally, if (V, *) is a (graded) commutative algebra, the quasi-shuffle (or overlapping shuffle, or stuffle) product □ on T'(V) is a deformation ([40], § 6) of the shuffle product characterized by the identity

(v • x)ul(w • y) = v • (x lu (w • y)) + w • ((v • y) + (v * w) • (xuiy) .

In particular, if we define an algebra structure on the graded vector space spanned by classes f dual to the Z(s by f *fj = f+j, we recover the quasi-shuffle product on the dual of NSymm*.

The Lie algebra of primitives in NSymm* is generated by the analogs of Newton's power functions ([23], § 4.1.F), and map under abelianization to the classical power function primitives in Symm*; dualizing yields a morphism of Symm* to QSymm*, which rationalizes to the asserted inclusion. □

2.3 Remarks

i) The applications below will be based on a variant of £* defined by generators in degree 4k +1, k > 0, rather than 2k +1. The corresponding free Lie algebras will then have generators in (homological) degree -2(2k + 1). This doubling of topological degree relative to motivic weight is a familiar consequence of differing conventions.

ii) Hoffman ([40], Theorem 2.5) constructs an isomorphism

exp : QSymm* < Q ^ QSymm* < Q

of graded Hopf algebras over the rationals, taking lu to lD ;so over Q we can think of the morphism defined by the proposition as the inclusion of the symmetric functions in the quasisymmetric functions with the quasishuffle product.

iii) The rationalization NSymm* < Q is the (primitively generated) universal enveloping algebra U(f*) of the free Lie algebra f* generated by the Z's over Q. By Poincare-Birkhoff-Witt its modules can be regarded as representations of a pro-unipotent groupscheme G0(f*) over Q, or equivalently as comodules over the Hopf algebra QSymm* < Q of algebraic functions on that pro-unipotent group. If we interpret graded modules as representations of the multiplicative groupscheme in the usual way ([3], § 3.2.7), see also ([8], § 1.1.2), then we can regard these modules as representations of a proalgebraic groupscheme

1 ^ G0(f*) ^ G(f*) := Gm k G0(f*) ^ Gm ^ 1.

In a very helpful appendix, Deligne and Goncharov ([27], § A.15) characterize representations of G(f*) as graded f*-modules, such that (if Q(n) denotes a copy of Q in degree n)

This is explained in more detail in ([35], § 8); we will return to this description below.

iv) The rational stable homotopy category is equivalent to the derived category of rational vector spaces, and the homotopy category of rational ring-spectra is equivalent to the homotopy category of DGAs: the Hurewicz map

[ S*, Xq] = nS(X) ® Q ^ H* (X, Q)

is an isomorphism. This leads to a convenient abuse of notation which may not distinguish the rationalization Xq of a spectrum from its homology (or its homotopy). For example,

(Q(0), Q(n)) = (4)v .

Rep(g(f*))

the rational de Rham algebra of forms on a reasonable space is a good model for the rational Spanier-Whitehead commutative ringspectrum [X, SQ].

2.4 Finite-dimensional graded modules over a field k have a good duality functor

V* ^ Homk(V*, k) = (V*)v ,

and a great deal of work on the homological algebra of augmented algebras $ : A ^ k (and their generalizations) is formulated in terms of constructions generalizing

M* ^ Homk(M* ®a B*(A/k),k) = HomA(M*,B*(A/k)v) := RHomA(M*,k) ,

where B* (A/k)v is now essentially a cobar construction. This is the classical contravariant Koszul duality functor: see ([12,21], [30], § 4.22) for recent work in the context of modules over ring spectra.

In this paper, however, we work instead (following [34,60]) with the covariant functor M* ^ Mj defined as in § 2.2.2, but regarded as mapping modules over an augmented algebra A to comodules over a coaugmented coalgebra Aj. In particular, in the case of our main example (over Q) Hess's hypotheses ([38], § 5.3) are statisfied, and we have the

Corollary. The Hess-Koszul-Moore functor

LqE : Db(E* <g> Q - Mod) ^ Db(QSymm* <g> Q - Comod) is an equivalence of symmetric monoidal categories.

Proof. The point is that, in the context of graded commutative augmented algebras A over a field k, the functor Lq* is monoidal, in the sense that

Lq* (Mo) <k Lq>* (Mi) := (Mo <A B(A/k)) <k k <k (Mi <a B(A/k)) <k k

is homotopy-equivalent to

Lq(Mo <a Mi) := ((Mo <A Mi) <a B(A/k)) <k k

via the morphism

(Mo <a B(A/k)) <k Mi ^ Mo <<a Mi

induced by the homotopy equivalence of B(A/k) with k as an A-module. This then lifts to an equivalence Lqj of comodules, cf [58,68].

In the terminology of § 2.2.3iii, the composition

PBW o exp* oLqE := L$>t

thus defines an equivalence of the derived category of E* < Q-modules with the derived category of G(f*)-representations. A similar argument identifies the bounded derived category of modules overE*<Q with the bounded derived category of representations of the graded abelianization G(f*b) of G(f*): in other words, of graded modules over Symm* < Q.

Remarks. The quite elementary results above were inspired by groundbreaking work of Baker and Richter [6], who showed that the integral cohomology of ^SCP^ is isomorphic to QSymm* as a Hopf algebra. Indeed, the £2-term

TorH*(Z) (H* (X),H*(Y)) ^ H* (X xZY)

of the Eilenberg-Moore spectral sequence for the fiber product

..........>P£CP~

is the homology of the bar construction on the algebra H* (SCP^), which is a square-zero extension of Z. The spectral sequence collapses for dimensional reasons but has nontrivial multiplicative extensions (connected to the fact that H*(CPTO) is polynomial (cf § 2.2.2, [5,50]).

In view of Proposition 3.2.1 below, Proposition 2.2.1 can be rephrased as the algebraically similar assertion that the Kunneth spectral sequence ([32], IV § 4.1)

TorH*(K(s0),Q)(H*(S0,Q),H*(S0,Q)) ^ H*(K(S0)f,Q)

for ring-spectra collapses. In this case the algebra structure on H*(K(S°),Q) is trivial, resulting in the shuffle algebra QSymm*. Note that although these spectral sequences look algebraically similar, one is concentrated in positive, the other in negative, degrees. If or how they might be related, e.g., via the cyclotomic trace (cf § 4.1), seems quite mysterious to the author.

I am deeply indebted to Baker and Richter for help with this, and with many other matters. I am similarly indebted to John Rognes for patient attempts to educate me about the issues in the section following.

3 Geometric generators for K*(S°) ® Q

3.1 Stable smooth cell bundles are classified by a space (i.e., simplicial set)

colim„^TO BDiff(Dn) ,

where Diff(Dn) is the group of diffeomorphisms of the closed «-disk (which are not required to fix the boundary3). Following ([72], § 1.2, § 6.1), there is a fibration

HDiff(Sn-1) ^ BDiff(Dn) ^ BDiff(D0n) ,

where D0n is the open disk, and HDin(Sn-1) is the simplicial set of smooth ^-cobordisms of a sphere with itself [71]. The homomorphism O(n) ^ Diff(D0n) is a homotopy equivalence, while the constructions of ([72], § 3.2) define a system

colimn^ BHDiff(Sn-1) ^ Wh(*) = ^TOK(S0)

of maps to the fiber of the Dennis trace, which becomes a homotopy equivalence in the limit. It follows that the K-theory groups

colimn^ ^*BDiff(Dn) := K*ceU

(of smooth cell bundles over a point) satisfy

KceU ® Q = Q2 if i = 4k > 0

and are zero for other positive i, cf e.g., ([42], p 7).

The resulting parallel manifestations of classical zeta-values in algebraic geometry, and in algebraic and differential topology, seem quite remarkable, and I am arguing here that they have a unified origin in the fibration

^Wh(*)-► BDiff(D)-► BO

with odd negative zeta-values originating in the /-homomorphism to Q(S°) on the right, and odd positive zeta-values originating in pseudoisotopy theory through K(S0) on the left. The adjoint functors B and ^ account for the shift of homological dimension by two, from K4k-1(Z) (where f(1 - 2k) lives) to K4k+1(Z) (where f(1 + 2k) lives).

One can hope that this provocative fact might someday provide a basis for a theory of smooth motives (conceivably involving the functional equation of the zeta-function), but at the moment, even the multiplicative structure of K£eU < Q is obscure to me.

3.2.1 Work of Rognes [63], sharpening earlier constructions of Hatcher ([36], § 6.4), Waldhausen, and Bokstedt, provides geometrically motivated generators for K(S0)* < Q by defining a rational infinite-loop equivalence

f : B(F/O) ^ Wh(*) .

Here F is the monoid of homotopy self-equivalences of the stable sphere [56]; I will write f/O for the spectrum defined by the infinite loopspace F/O. One of my many debts to this paper's referees is the construction of a rational equivalence

(S0 v SkO) < Q ^ (S0 v Sf/O) < Q

of ring-spectra (with simple multiplication) via the zigzag

BO = * xO £O «-F xO £O-► F xO * = F/O

of maps of infinite loopspaces; together with Rognes's construction, this defines an equivalence

w : (S0 v SkO) < Q ^ K(S0) < Q

of rational ring-spectra (alternately: of DGAs with trivial differentials and product structure).

Proposition. The resulting homomorphism w* : Q 0 kO*[1] <Q ^ K*(S0) < Q presents the rationalization of K(S°) as a square-zero extension of Q by an ideal Q{a vk | k > 1}

(| v| = 4) with trivial multiplication. □

3.2.2 Writing S0[ X+] for the suspension spectrum of a space X emphasizes the similarity of that construction to the free abelian group generated by a set. The equivalence

MapsSpaces(iix'Z+, Z+) = MapsSpectra(S°[ fi~Z+],Z)

sends the identity map on the left to a stabilization morphism

S0[ n™Z+] ^ Z

of spectra: for example, if Z = XkO then is the Bott space SU/SO, and the extension

S0[SU/SO+] ^ S0 v XkO

of stabilization by the collapse map SU/SO ^ S0 to a map of ring-spectra (with the target regarded as a square-zero extension) is the product-killing quotient

e4k+1 ^ avk : H* (SU/SO, Q) = E (e4k+1 | k > 1) ® Q ^ Q 0 Q {avk | k > 1} .

3.2.3 The Kunneth spectral sequence

ToH(SU/SO,Q) (h* (s0, Q), h* (S0, Q)) ^ H* ((S0 [ SU/SO+]f, Q)

([32], IV § 4.1) for the rational homology of S0 A50[SU/SO+] S0 collapses, yielding an isomorphism of its target with the algebra of symmetric functions on generators of degree 4k+2, k > 0. It is algebraically isomorphic to the Rothenberg-Steenrod spectral sequence

TorH*(SU/SO,Q) (Q, Q)) ^ H* (Sp/SU,«

SQ[Sp/SU + ]

([59], § 7.4) forB(SU/SO), allowing us to identifySQ[Sp/SU, ] withthe covariant Koszul

dual of SQ[SU/SO+]. The composition

SQ[Sp/SU+] = (S0 AS0[SU/SO+] S0)Q ^ (S0 As0vXkO S0)Q ^ (S0 ak(s0) S)Q = K(S0)Q represents the abelianization map

G(f*) ^ G(fth) (= Spec H*(Sp/SU, Q)) of § 2.4 above.

3.2.4 Remarks

1) v2 is twice the Bott periodicity class.

2) The arguments above are based on the equivalence, over the rationals, of K(S°) and K(Z). In a way this is analogous to the isomorphism between singular (Betti) and algebraic de Rham (Grothendieck) cohomology of algebraic varieties. Nori ([48], Theorem 6) formulates the theory of periods in terms of functions on the torsor of isomorphisms between these theories; from this point of view, zeta-values appear as functions on Spec (K(Z)*K(S0)), viewed as a torsor relating arithmetic geometry to differential topology.

3.3 The Tannakian category of mixed Tate motives over Z constructed by Deligne and Goncharov is equivalent to the category of linear representations of the motivic group GTMT of that category (thought to be closely related to Drinfel'd's prounipotent version of the Grothendieck-Teichmuller group ([2], § 25.9.4; [28]; [73] § 6.1, Prop 9.1)). At the end of a later paper, Goncharov describes the Hopf algebra Hgtmt) of functions on this motivic group in some detail: in particular ([35], § 8.2 Theorem 8.2, § 8.4 exp (110)) he identifies it as the cotensor algebra T'(KqZ), where

KqZ := 0n>1K2n-1 (Z) ® Q (regarded as a graded module with K2n-1 situated in degree n).

The composition of the pseudo-isotopy map w* of § 3.2.1 with Waldhausen's isomorphism K(S°) < Q = K(Z) < Q identifies the free graded Lie algebra on KqZ with the free Lie algebra f* of § 2.3iii above, yielding an isomorphism

G(f*) ^ GTmt

of proalgebraic groups. Corollary 2.4 then implies the

Theorem. The composition

w* o L$f : Db(K*(S0) < Q - Mod) ^ Db(GTmt - Mod) ,

defines an equivalence of the homotopy category of rational K(S0)-module spectra with the derived category of mixed Tate motives over Z. □

4 Some applications

This section discusses some applications of the preceding discussion. The first paragraph below is essentially an acknowledgement of ignorance about topological cyclic homology. The second discusses some joint work in progress [47] with Nitu Kitchloo. The setup and ideas are entirely his; the section below sketches how Koszul duality seems to fit in with them. I am indebted to Kitchloo for generously sharing these ideas with me.

The third paragraph summarizes some of the work of Blumberg, Gepner, and Tabuada mentioned in the 'Introduction', concerned with a program for constructing enriched decategorifications of their approach to generalized motives as small stable ^-categories.

4.1 Topological cyclic homology ([13,17], ...) is a powerful tool for the study of the algebraic K-theory of spaces, and its role in these matters deserves discussion here; but at the moment, there are technical obstructions to telling a coherent story. The current state of the art defines local invariants TC(X;p) for a space at each primep (closely related to the homotopy quotient of the suspension of the free loopspace of X), whereas the theory of mixed Tate motives over integer rings is intrinsically global. For example, the topological cyclic homology of a point looks much like the p-completion of an ad hoc geometric model

TCgeo(S0) - S0 v XCP~

([9,57], [62], § 3) with

H* (TCgeo(S0), Z) = Z 0 Z{atk | k > -1} .

The (rational) Koszul dual of this object defines a proalgebraic groupscheme associated to a free graded Lie algebra roughly twice as big as f*, i.e., with generators in topological degree -2k rather than -2(2k + 1). A similar group appears in work of Connes and Marcolli ([25], Prop 5.4) on renormalization theory, and topological cyclic homology is plausibly quite relevant to that work; but because the global arithmetic properties of topological cyclic homology are not yet well understood, it seems premature to speculate further here; this remark is included only to signal this possible connection to physics.

4.2 Example: Kitchloo [46] has defined a rigid monoidal category sS with symplectic manifolds (M, a) as objects, and stable equivalence classes of oriented Lagrangian correspondences as morphisms. It has a fiber functor which sends such a manifold (endowed with a compatible almost-complex structure) to a Thom spectrum

sQ(M) = U/SO(Tm)-z

constructed from the U/SO-bundle of Lagrangian structures on its stable tangent space. An Eilenberg-Moore spectral sequence with

E2 = Torf {BU)(H*(M), H *(BSO))

([59], § 7.4 ) computes H*(U/SO(TM)), and away from the prime two, the equivariant Borel cohomology

H^/So(sQ(M)) := H*(sQ(M) xu/so E(U/SO))

is naturally isomorphic to H*(M).

The functor sQ(-) has many of the formal properties of a homology theory; for example, when M is a point, sQ := sQ(*) is a ring-spectrum [65], and sQ(-) takes values in the category of sQ-modules. Moreover, when V is compact oriented, with the usual symplectic structure on its cotangent space,

sQ(T*V) - [ V,sQ]

([46], § 2.6) defines a cobordism theory of Lagrangian maps (in the sense of Arnol'd) to V. The composition

sQ(M) ^ sQ(M) AM+ ^ sQ(M) A B(U/SO)+

(defined by the map M ^ B(U/SO) which classifies the bundle U/SO(TM) of Lagrangian frames on M) makes sQ(M) a comodule over the Hopf spectrum

THH (sQ) - sQ A B(U/SO)+ - sQ[Sp/U+]

(the analog, in this context, of an action of the abelianization G(f*b) ([47], § 4)). The Hopf algebra counit

1:SQ[U/SO+] ^ SQ

e H° (U/SO,(

provides, via the Thom isomorphism, an augmentation sQq ^ SQ] e H *(sQ, Q) - H * (U/SO, Q) .

Proposition. The covariant Koszul dual

sQ(M)Q := sQ(M) aQ sQ is a comodule over

SQ asQq SQ - SQ AS° [u/so+] SQ - S° [Sp/U+] ;

by naturality its contravariant Koszul dual

RHomsQQ(SQ,sQQ(M)) - sQq(M)

inherits an sQQ[Sp/U+] - coaction: equivalently, an action of the abelianized Grothendieck-Teichmuller group G(f*b).

Remarks. i) It seems likely that this coaction agrees with the THH(sQ)-coaction described above.

ii) If M = T* V is a cotangent bundle, we have an isomorphism

sQQ(M) = H*(V,Q) .

iii) The sketch above is proposed as an analog, in the theory of geometric quantization, to work ([48], § 4.6.2, § 8.4) of Kontsevich on deformation quantization. A version of the Grothendieck-Teichmuller group acts on the Hochschild cohomology

HHC(M) := Ext^MxM (Om, OM) = 0 H*(M, A*TM))

of a complex manifold (defined in terms of coherent sheaves of holomorphic functions on M x M). If M is Calabi-Yau, its tangent and cotangent bundles can be identified, resulting in an action of the abelianized Grothendieck-Teichmuller group on the Hodge cohomology of M.

Note that Sp/U ~ BT x Sp/SU splits. The action on sQ of the two-dimensional cohomology class carried by BT does not seem to come from a K(S°)^ coaction but rather from variation of the symplectic structure. This may be related to Kontsevich's remarks (just after Theorems 7 and 9) about Euler's constant.

4.3 Example Marshaling the forces of higher category theory, Blumberg, Gepner, and Tabuada [10] have developed a beautiful approach to the study of noncommutative motives, defining symmetric monoidal categories M (there are several interesting variants ([10], § 6.7, § 8.10)) whose objects are small stable ^-categories (e.g., of perfect complexes of quasicoherent sheaves of modules over a scheme, or of suitably small modules over the Spanier-Whitehead dual ring-spectrum [X, S0] of a finite complex). The morphism objects

Mor m(A, B)

in these constructions are K-theory spectra of categories of exact functors between A and B; this defines spectral enrichments over the homotopy category of K(S0)-modules ([11], Corollary 1.11).

The arguments of this paper imply that covariant Koszul duality, as outlined above, defines versions of these categories with morphism objects

MorHo(m)(A,B)Q e K(S0)Q - Comod

which, under suitable finiteness conditions, may be regarded as enriched over ^¿(MTq (Z)). They suggest the existence of categories Ho(M) with morphism objects

MorHo(M)(A, B) e K(S0)t - Comod

which rationalize to the categories described above. This seems to fit well with recent work [66] on Konsevich's conjecture on noncommutative motives over a field ([61], § 4.4). The theory of cyclotomic spectra [13] suggests the existence of related constructions from that point of view, but (as noted in § 4.1) their arithmetic properties are not yet very well understood.

Recently F Brown, using earlier work of Zagier [74], has shown that the algebra %Gtmt is isomorphic to a polynomial algebra

Q[ Zm(w) | w e Lyndon{2,3}] := Q[ Zm]

of motivic polyzeta values indexed by certain Lyndon words (cf ([20], § 3 exp 3.6, § 8): working with motivic polyzetas avoids questions of algebraic independence of numerical polyzetas). This suggests the category Ho(m)q[zmP with morphism objects

MorHo(m)(a, b)q 0gtmi Q[ zm] as a convenient 'untwisted' Q[ Z m]-linear category of noncommutative motives.

Endnote

aThis paper was inspired by Graeme Segal's description of such objects as 'blancmanges'.

Acknowledgements

1 am deeply indebted to Andrew Blumberg, Kathryn Hess, and Nitu Kitchloo for help and encouragement in the early stages of this work; and to Andrew Baker, Birgit Richter, and John Rognes for their advice and intervention in its later stages. Thanks to all of them - and to some very perceptive and helpful referees - for their interest and patience. Any mistakes, misunderstandings, and oversimplifications are my responsibility.

Received: 14 March 2014 Accepted: 17 April 2015 Published online: 10 June 2015

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